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Properties

Label 322.3.d.a
Level $322$
Weight $3$
Character orbit 322.d
Analytic conductor $8.774$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,3,Mod(183,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.183");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 322.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.77386451240\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{3} + 48 q^{4} + 112 q^{9} + 16 q^{12} + 8 q^{13} + 96 q^{16} + 32 q^{18} - 24 q^{23} + 24 q^{25} - 32 q^{26} + 32 q^{27} - 168 q^{29} - 208 q^{31} - 56 q^{35} + 224 q^{36} + 152 q^{39} + 40 q^{41}+ \cdots - 416 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
183.1 −1.41421 −4.61520 2.00000 − 3.27883i 6.52688 2.64575i −2.82843 12.3001 4.63696i
183.2 −1.41421 −4.61520 2.00000 3.27883i 6.52688 − 2.64575i −2.82843 12.3001 − 4.63696i
183.3 −1.41421 −3.76306 2.00000 − 2.49334i 5.32177 2.64575i −2.82843 5.16060 3.52611i
183.4 −1.41421 −3.76306 2.00000 2.49334i 5.32177 − 2.64575i −2.82843 5.16060 − 3.52611i
183.5 −1.41421 0.385477 2.00000 − 5.05911i −0.545147 − 2.64575i −2.82843 −8.85141 7.15466i
183.6 −1.41421 0.385477 2.00000 5.05911i −0.545147 2.64575i −2.82843 −8.85141 − 7.15466i
183.7 −1.41421 1.48456 2.00000 − 2.70409i −2.09948 2.64575i −2.82843 −6.79610 3.82416i
183.8 −1.41421 1.48456 2.00000 2.70409i −2.09948 − 2.64575i −2.82843 −6.79610 − 3.82416i
183.9 −1.41421 3.17362 2.00000 − 5.33493i −4.48817 2.64575i −2.82843 1.07184 7.54473i
183.10 −1.41421 3.17362 2.00000 5.33493i −4.48817 − 2.64575i −2.82843 1.07184 − 7.54473i
183.11 −1.41421 5.33461 2.00000 − 6.56026i −7.54428 − 2.64575i −2.82843 19.4581 9.27761i
183.12 −1.41421 5.33461 2.00000 6.56026i −7.54428 2.64575i −2.82843 19.4581 − 9.27761i
183.13 1.41421 −4.30063 2.00000 − 1.54005i −6.08201 2.64575i 2.82843 9.49543 − 2.17796i
183.14 1.41421 −4.30063 2.00000 1.54005i −6.08201 − 2.64575i 2.82843 9.49543 2.17796i
183.15 1.41421 −4.29238 2.00000 − 7.91841i −6.07034 − 2.64575i 2.82843 9.42453 − 11.1983i
183.16 1.41421 −4.29238 2.00000 7.91841i −6.07034 2.64575i 2.82843 9.42453 11.1983i
183.17 1.41421 −1.26933 2.00000 − 2.74802i −1.79510 − 2.64575i 2.82843 −7.38880 − 3.88628i
183.18 1.41421 −1.26933 2.00000 2.74802i −1.79510 2.64575i 2.82843 −7.38880 3.88628i
183.19 1.41421 3.32026 2.00000 − 8.50314i 4.69556 − 2.64575i 2.82843 2.02413 − 12.0252i
183.20 1.41421 3.32026 2.00000 8.50314i 4.69556 2.64575i 2.82843 2.02413 12.0252i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 183.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.3.d.a ✓ 24
23.b odd 2 1 inner 322.3.d.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.3.d.a ✓ 24 1.a even 1 1 trivial
322.3.d.a ✓ 24 23.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).